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One simplification maybe useful is to image some situation where the coupling saving can be omit and only the statistical effect of Pauli exclusion matters.  This would be the situation where the coupling is infinitesimal small but we can wait infinite long time.  So the two channels are only related by the equal chemical potential.  If there is no common species, then there is no Pauli exclusion and all paricles goes into one channel until the chemical potential is higher than the other, while the only thing moves the chemical potential is the Fermi statistics within the channel.  On the case where there is a common specie, the chemical potential is afftected not only by the density within the species but also by that of the other species.

What happens to two-body two-channel cross-element of T-matrix?  0? or 1? 

It might be a good idea to take the coupling to zero in the variation approach, which seems to be along the long of the above idea. 